Variant of Abby and Bob and 3 numbers on the blackboard

Question

Abby selects a positive integer $A$, and Bobby selects a positive integer $B$. They then both tell their number secretly to Summy. Summy writes the numbers $5$, $8$, and $15$ on the blackboard, and announces that one of these three numbers is the sum $A+B$. Then the three go through several rounds of the following form:

• Summy rings a bell.
• Abby writes on a slip of paper whether she does know or does not know which of the numbers on the blackboard is the sum $A+B$.
• Bobby writes on a (different) slip of paper whether he does know or does not know which of the numbers on the blackboard is the sum $A+B$.
• Abby and Bobby give their papers to Summy.
• Summy checks the papers. If at least one of the papers says YES, then the process stops. If both papers say NO, then the next round starts.

Abby and Bobby are absolutely honest and very very intelligent.

Question: What is the maximum number of times that the bell might be rung before the process stops?

This is a variant of the following puzzle:

Abby and Bobby and three numbers on the blackboard

The difference is that in the original puzzle, 0 could also be selected by Abby and Bob. In this variant, 0 cannot be selected.

Answer 1

4 rings

The first ring happens before any information is passed, so we can't draw any conclusions

If the second ring happens,

both players can assume A<8 and B<8, as otherwise the sum to 15 is obvious

If the third ring happens,

both players can assume A<5 and B<5, as otherwise the sum to 8 is obvious, given the A<8, B<8 constraint from the prior step.

If the fourth ring happens,

4 is the only number capable of summing to both 5 and 8, given the A<5, B<5 constraint.